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Logarithmic Differentiation Practice Problems

Take ln of both sides, expand with the log rules, differentiate to get y′/y, then multiply by y. Leave your answer in terms of x only.

  1. 1.

    Differentiate . Type ln as ln( ), for example x^x*(ln(x) + 1).

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    , so and .

  2. 2.

    Differentiate . Put a longer exponent in brackets: x^(2x).

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    , so and .

  3. 3.

    Use logarithmic differentiation to differentiate .

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    , so . Multiplying out gives , the same as expanding first; either form is accepted.

  4. 4.

    Differentiate .

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    , so and .

  5. 5.

    Differentiate .

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    , so and .

  6. 6.

    Differentiate .

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    . The quotient rule gives , so .

  7. 7.

    Differentiate .

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    , so .

  8. 8.

    Differentiate . Type square roots as sqrt( ).

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    , so .

  9. 9.

    Differentiate .

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    , so and .

  10. 10.

    Differentiate , for .

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    . The product rule, with the chain rule on , gives . So .

  11. 11.

    Find the slope of the tangent line to at .

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    . At : .

  12. 12.

    For , find .

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    , so . At : and , so .
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MPhil in Mathematics, Founder of Mathovia
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Logarithmic Differentiation Practice Problems

These problems follow the logarithmic differentiation lesson. They start with and build up to variable powers and long products and quotients.

The method

  1. Take of both sides of .
  2. Expand with the log rules: products become sums, quotients become differences, powers become multiples.
  3. Differentiate both sides: the left side becomes .
  4. Multiply by and replace with the original function.

Common mistakes to watch for

  • Stopping at without multiplying by .
  • Splitting , which has no log rule.
  • Using the power rule on , which gives only half the answer.
  • Leaving in the final answer.

Frequently Asked Questions

How should I type my answer?+

Write it in terms of x only, replacing y with the original function. Use ln( ) for the natural log and put exponents longer than one symbol in brackets, for example x^(sin(x))*(cos(x)*ln(x) + sin(x)/x).

Do I need to multiply out the bracket?+

No. The usual form, the original function times a bracket of simple terms, is accepted, and so is any expanded or simplified version that is equal to it.

More practice problems

Implicit Differentiation Practice Problems Derivatives of Exponential and Logarithmic Functions Practice Problems Product Rule Practice Problems